Open Access
November 2012 Structured Sparsity through Convex Optimization
Francis Bach, Rodolphe Jenatton, Julien Mairal, Guillaume Obozinski
Statist. Sci. 27(4): 450-468 (November 2012). DOI: 10.1214/12-STS394


Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the regularization by the $\ell_{1}$-norm. In this paper, we consider situations where we are not only interested in sparsity, but where some structural prior knowledge is available as well. We show that the $\ell_{1}$-norm can then be extended to structured norms built on either disjoint or overlapping groups of variables, leading to a flexible framework that can deal with various structures. We present applications to unsupervised learning, for structured sparse principal component analysis and hierarchical dictionary learning, and to supervised learning in the context of nonlinear variable selection.


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Francis Bach. Rodolphe Jenatton. Julien Mairal. Guillaume Obozinski. "Structured Sparsity through Convex Optimization." Statist. Sci. 27 (4) 450 - 468, November 2012.


Published: November 2012
First available in Project Euclid: 21 December 2012

zbMATH: 1331.90050
MathSciNet: MR3025128
Digital Object Identifier: 10.1214/12-STS394

Keywords: Convex optimization , Sparsity

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.27 • No. 4 • November 2012
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